Marc E. Pfetsch

The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing

This paper deals with the computational complexity of conditions which guarantee that the NP-hard problem of finding the sparsest solution to an underdetermined linear system can be solved by efficient algorithms. In the literature, several such conditions have been introduced. The most well-known ones are the mutual coherence, the restricted isometry property (RIP), and the nullspace property (NSP). While evaluating the mutual coherence of a given matrix is easy, it has been suspected for some time that evaluating RIP and NSP is computationally intractable in general. We confirm these conjectures by showing that for a given matrix A and positive integer k, computing the best constants for which the RIP or NSP hold is, in general, NP-hard. These results are based on the fact that determining the spark of a matrix is NP-hard, which is also established in this paper. Furthermore, we also give several complexity statements about problems related to the above concepts.

A Column-Generation Approach to Line Planning in Public Transport

The line-planning problem is one of the fundamental problems in strategic planning of public and rail transport. It involves finding lines and corresponding frequencies in a transport network such that a given travel demand can be satisfied. There are (at least) two objectives: the transport company wishes to minimize operating costs, and the passengers want to minimize traveling times. We propose a new multicommodity flow model for line planning. Its main features, in comparison to existing models, are that the passenger paths can be freely routed and lines are generated dynamically. We discuss properties of this model, investigate its complexity, and present a column-generation algorithm for its solution. Computational results with data for the city of Potsdam, Germany, are reported.

Models for fare planning in public transport

The optimization of fare systems in public transit allows to pursue objectives such as the maximization of demand, revenue, profit, or social welfare. We propose a nonlinear optimization approach to fare planning that is based on a detailed discrete choice model of user behavior. The approach allows to analyze different fare structures, optimization objectives, and operational scenarios involving, e.g., subsidies. We use the resulting models to compute optimized fare systems for the city of Potsdam, Germany.

On the Maximum Feasible Subsystem Problem, IISs and IIS-hypergraphs

Abstract. We consider the Max FS problem: For a given infeasible linear system Ax≤b, determine a feasible subsystem containing as many inequalities as possible. This problem, which is NP-hard and also difficult to approximate, has a number of interesting applications in a wide range of fields. In this paper we examine structural and algorithmic properties of Max FS and of Irreducible Infeasible Subsystems (IISs), which are intrinsically related since one must delete at least one constraint from each IIS to attain feasibility. First we provide a new simplex decomposition characterization of IISs and prove that finding a smallest cardinality IIS is very difficult to approximate. Then we discuss structural properties of IIS-hypergraphs, i.e., hypergraphs in which each edge corresponds to an IIS, and show that recognizing IIS-hypergraphs subsumes the Steinitz problem for polytopes and hence is NP-hard. Finally we investigate rank facets of the Feasible Subsystem polytope whose vertices are incidence vectors of feasible subsystems of a given infeasible system. In particular, using the IIS-hypergraph structural result, we show that only two very specific types of rank inequalities induced by generalized antiwebs (which generalize cliques, odd holes and antiholes to general independence systems) can arise as facets.

Validation of nominations in gas network optimization: models, methods, and solutions

In this article, we investigate methods to solve a fundamental task in gas transportation, namely the validation of nomination problem: given a gas transmission network consisting of passive pipelines and active, controllable elements and given an amount of gas at every entry and exit point of the network, find operational settings for all active elements such that there exists a network state meeting all physical, technical, and legal constraints. We describe a two-stage approach to solve the resulting complex and numerically difficult nonconvex mixedinteger nonlinear feasibility problem. The first phase consists of four distinct algorithms applying mixedinteger linear, mixedinteger nonlinear, nonlinear, and methods for complementarity constraints to compute possible settings for the discrete decisions. The second phase employs a precise continuous nonlinear programming model of the gas network. Using this setup, we are able to compute high-quality solutions to real-world industrial instances that are significantly larger than networks that have appeared in the mathematical programming literature before.

Computing Optimal Morse Matchings

Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is \NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results.

Orbitopal Fixing

The topic of this paper are integer programming models in which a subset of 0/1-variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branch-and-cut algorithms if the order of the subsets of the partition is irrelevant. This kind of symmetry unnecessarily blows up the branch-and-cut tree. We present a general tool, called orbitopal fixing, for enhancing the capabilities of branch-and-cut algorithms in solving such symmetric integer programming models. We devise a linear time algorithm that, applied at each node of the branch-and-cut tree, removes redundant parts of the tree produced by the above mentioned symmetry. The method relies on certain polyhedra, called orbitopes, which have been investigated in [11]. It does, however, not add inequalities to the model, and thus, it does not increase the difficulty of solving the linear programming relaxations. We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem motivated from frequency planning in mobile telecommunication networks.

Evaluating Gas Network Capacities

This book addresses a seemingly simple question: Can a certain amount of gas be transported within a pipeline network? The question is difficult, however, when asked in relation to a meshed nationwide gas transportation network and when taking into account technical details and discrete decisions, as well as regulations, contracts, and varying demands involved. Evaluating Gas Network Capacities provides an introduction to the field of gas transportation planning and discusses in detail the advantages and disadvantages of several mathematical models that address gas transport within the context of the technical and regulatory framework. It shows how to solve the models using sophisticated mathematical optimization algorithms and includes examples of large-scale applications of mathematical optimization to this real-world industrial problem. Readers will also find a glossary of gas transport terms, tables listing the physical and technical quantities and constants used throughout the book, and a reference list of regulation and gas business literature. Audience: This book is intended for mathematicians interested in industrial applications. Engineers working in gas transport will also find the book of interest.

Some Algorithmic Problems in Polytope Theory

Many interesting algorithmic problems naturally arise in the theory of convex polytopes. In this article we collect 35 such problems and briefly discuss the current knowledge on their complexity status.

Computing the face lattice of a polytope from its vertex-facet incidences

We give an algorithm that constructs the Hasse diagram of the face lattice of a convex polytope P from its vertex-facet incidences in time O(min{n, m}.α.φ), where n is the number of vertices, m is the number of facets,α is the number of vertex-facet incidences, and φ is the total number of faces of P. This improves results of Fukuda and Rosta [Computational Geometry 4 (4) (1994) 191-198], who described an algorithm for enumerating all faces of a d-polytope in O(min{n, m} . d. φ2) steps. For simple or simplicial d-polytopes our algorithm can be specialized to run in time O(d. α. φ). Furthermore, applications of the algorithm to other atomic lattices are discussed, e.g., to face lattices of oriented matroids.